#7045364

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1293 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,24)(19,29,30,41)(20,26,34,39)(21,28,23,27)(25,32,38,33)(31,45,35,48)(36,44,37,43)(40,47,42,46);(1,3,5,8)(2,6,4,7)(9,11,13,10)(12,14,16,15)(18,24)(19,30)(20,34)(27,28)(32,33)(36,37)(40,42)(45,48);(1,2,5,4)(3,6,8,7)(10,11)(12,16)(17,19,21,32,22,30,23,33)(18,25,27,41,24,38,28,29)(20,35,37,46,34,31,36,47)(26,42,44,45,39,40,43,48);(1,4,5,2)(3,6,8,7)(9,12,13,16)(10,14,11,15)(17,20,22,34)(18,26,24,39)(19,31,30,35)(21,37,23,36)(25,40,38,42)(27,44,28,43)(29,45,41,48)(32,47,33,46);(10,11)(14,15)(17,21,22,23)(18,27,24,28)(19,32,30,33)(20,36,34,37)(25,41,38,29)(26,43,39,44)(31,46,35,47)(40,45,42,48)

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,c^2=fg,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6952560);